The parent state-specific multireference coupled-cluster (SS-MRCC) theory proposed by Mukherjee et al. [J. Chem. Phys. 110, 6171 (1999)], though rigorously size-extensive and also size-consistent with localized orbitals, has some deficiencies in the minimal truncation scheme, viz. at the singles and doubles (SD) level (SS-MRCCSD). SS-MRCCSD does not involve the direct coupling of all the model functions with a given virtual function belonging to the uncontracted multiconfiguration CISD space. It also does not involve, even in the linear power of a cluster operator T(μ), the direct coupling of the virtual functions χ(l(μ)), which are up to doubly excited with respect to a model function φ(μ) to the other virtual functions of the MRCISD space which can be generated by triple and quadruple excitations from φ(μ). We argue that inclusion of a selection of triples and quadruples involving at most two inactive orbital excitations from every φ(μ) would ameliorate the shortcoming of the incomplete coupling of the triply and quadruply excited virtual functions which can couple with the singly and doubly excited ones. This extended ansatz for our SS-MRCC theory, to be called SS-MRCCSDtq by us, would still miss the direct coupling of the manifold of the model functions {φ(λ),λ ≠ μ} to singly and doubly excited virtual functions. However, this effect is expected to be less significant than the lack of the more complete virtual space couplings, these functions being many more numerous, suggesting the new methods to be significantly improved schemes. Excellent results on the potential energy surfaces of small molecules involving single, double, and triple bond dissociation bear out our expectations fully.